$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$
Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that $S$ is finite and find an n so large that $S_n$ approximate $S$ to three decimal places.

Solution: first of all, I think that we Will use L'hopital rule and then use root test while starting to solve this. But how?